Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{-7}{21n - 35} \div \dfrac{7n}{n(3n - 5)} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-7}{21n - 35} \times \dfrac{n(3n - 5)}{7n} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ -7 \times n(3n - 5) } { (21n - 35) \times 7n } $ $ p = \dfrac {-7 \times n(3n - 5)} {7n \times 7(3n - 5)} $ $ p = \dfrac{-7n(3n - 5)}{49n(3n - 5)} $ We can cancel the $3n - 5$ so long as $3n - 5 \neq 0$ Therefore $n \neq \dfrac{5}{3}$ $p = \dfrac{-7n \cancel{(3n - 5})}{49n \cancel{(3n - 5)}} = -\dfrac{7n}{49n} = -\dfrac{1}{7} $